Passing next to the condition p = 0, we see from (24). by considering the coefficients of sin x, cos as, that ~ ~Ji + 9a/ + terms of 3rd order = 0, clt -7- + ga 4- terms of 3rd order = 0. . (jib The coefficients of sin 2#, cos 2#, require, as in (14), (15), that 1 ,7/Q 1 /7/Q' r/2 _ nz jf J. UiKJ , 7 J. U/AJ U/ It b=--T7 — aa, o = ---- ^r + ~~o — g at g dt 2 Again, the coefficients of sin 3x, cos 3*, give 3a' ,0. q , ,„,, ~ 8 (a ~da}) ........... >( } c = - + |(a - a>) + dt 2 8 fSa'a-a^ (32) 8 Cda a) .......... ( } When /3, /3', 7, 7', vanish, these results are much simplified. We have 6' = -att', & = |(a'2-tta), ..................... (33) c' = _ 3a-' (a'a _ 3a«), c = _ ^ (3a'2 - a2) ............. (34) o o If the principal terms represent a purely progressive wave, we may take, as in (17), a = A cos nt, a' ••= A sin nt, ..................... (35) where n is for the moment undetermined. Accordingly 6/ = -i^lasin2n*, -b = - ^A2 cos 2n£, c' = M8 sin 87i«, c = | A3 cos 3?itf ; so that • y = A cos (a? - nt) - %AZ cos 2 (as - nt) + f J.8 cos 3 (x - nt), ...... (36) representing a progressive wave of permanent type, as found by Stokes. To determine n we utilize (28), (29), in the small terms of which we may take « = <7 a'dt = — ,— cos nt, of = *- g I adt = — — sin nt, so that * a?+az = A*n2. ' mi #( Thus . -J and %2n above the mean level exceeds numerically the maximum depression below it.